Trigonometry Examples

$f (x) = \cos (x + 2) + 1$

Step 1

Write $f (x) = \cos (x + 2) + 1$ as an equation.

$y = \cos (x + 2) + 1$

Step 2

Interchange the variables.

$x = \cos (y + 2) + 1$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\cos (y + 2) + 1 = x$ .

$\cos (y + 2) + 1 = x$

Step 3.2

Subtract $1$ from both sides of the equation.

$\cos (y + 2) = x - 1$

Step 3.3

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$y + 2 = \arccos (x - 1)$

Step 3.4

Subtract $2$ from both sides of the equation.

$y = \arccos (x - 1) - 2$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \arccos (x - 1) - 2$

Step 5

Verify if $f^{- 1} (x) = \arccos (x - 1) - 2$ is the inverse of $f (x) = \cos (x + 2) + 1$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (\cos (x + 2) + 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\cos (x + 2) + 1) = \arccos ((\cos (x + 2) + 1) - 1) - 2$

Step 5.2.3

Combine the opposite terms in $\arccos ((\cos (x + 2) + 1) - 1) - 2$ .

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Step 5.2.3.1

Subtract $1$ from $1$ .

$f^{- 1} (\cos (x + 2) + 1) = \arccos (\cos (x + 2) + 0) - 2$

Step 5.2.3.2

Add $\cos (x + 2)$ and $0$ .

$f^{- 1} (\cos (x + 2) + 1) = \arccos (\cos (x + 2)) - 2$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\arccos (x - 1) - 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arccos (x - 1) - 2) = \cos ((\arccos (x - 1) - 2) + 2) + 1$

Step 5.3.3

Combine the opposite terms in $\cos ((\arccos (x - 1) - 2) + 2) + 1$ .

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Step 5.3.3.1

Add $- 2$ and $2$ .

$f (\arccos (x - 1) - 2) = \cos (\arccos (x - 1) + 0) + 1$

Step 5.3.3.2

Add $\arccos (x - 1)$ and $0$ .

$f (\arccos (x - 1) - 2) = \cos (\arccos (x - 1)) + 1$

Step 5.3.4

The functions cosine and arccosine are inverses.

$f (\arccos (x - 1) - 2) = x - 1 + 1$

Step 5.3.5

Combine the opposite terms in $x - 1 + 1$ .

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Step 5.3.5.1

Add $- 1$ and $1$ .

$f (\arccos (x - 1) - 2) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (\arccos (x - 1) - 2) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arccos (x - 1) - 2$ is the inverse of $f (x) = \cos (x + 2) + 1$ .

$f^{- 1} (x) = \arccos (x - 1) - 2$